New cases of the Strong Stanley Conjecture

TL;DR

This paper advances the understanding of Littlewood-Richardson coefficients for Jack symmetric functions by proving new cases of Stanley's conjecture involving hook length formulas, especially for rectangular unions.

Contribution

It proves new cases of Stanley's conjecture for Littlewood-Richardson coefficients of Jack functions, including explicit formulas for rectangular unions and proposes a shifted analogue.

Findings

Proved Stanley's conjecture for rectangular union cases.

Derived explicit hook length formulas for these coefficients.

Conjectured a shifted Jack functions analogue of the conjecture.

Abstract

We make progress towards understanding the structure of Littlewood-Richardson coefficients $g_{\lambda,\mu}^{\nu}$ for products of Jack symmetric functions. Building on recent results of the second author, we are able to prove new cases of a conjecture of Stanley in which certain families of these coefficients can be expressed as a product of upper or lower hook lengths for every box in each of the partitions. In particular, we prove that conjecture in the case of a rectangular union, i.e. for $g_{\mu,\bar \sigma}^{\mu \cup m^n}$ where $\bar \sigma$ is the complementary partition of $\sigma = \mu \cap m^n$ in the rectangular partition $m^n$. We give a formula for these coefficients through an explicit prescription of such choices of hooks. Lastly, we conjecture an analogue of this conjecture of Stanley holds in the case of Shifted Jack functions.